Managing Credit Risk in Corporate Bond Portfolios : A Practitioner's ...

Mathematical Preliminaries 17The elements of the matrix L that represents the Cholesky decompositionof the matrix can be computed using the following rule:i1l ii a s ii B a lik 2 b ,l ji 1 i1a sl ji a l jk l ik b ,iik1k1i 1, 2, ... , nj i 1, ... , nI mentioned that covariance matrices estimated from historical data couldbe singular. If this happens, we artificially add some variance to each of therandom variables so that the covariance matrix is positive definite. Forinstance, if E denotes a diagonal matrix with small positive elements, thenthe matrix E has the property that it is positive definite and theCholesky decomposition can be computed.Markov MatrixA real n n matrix P [p ij ] is called a Markov matrix if its elements havethe following properties:na p ij 1, i 1,2, ... ,nj1p ij 0, i, j 1,2, ... ,nThis definition indicates that the elements in each row of a Markov matrixare non-negative and sum to one. As a result, any row vector having thisproperty can be considered to represent a valid probability mass function.This leads to the interpretation of any vector having this property as aprobability vector.Markov matrices have some interesting properties. The matrix formedby taking the product of two Markov matrices is also a Markov matrix. Ifone multiplies a probability vector by a Markov matrix, the result is anotherprobability vector. Markov matrices find applications in many differentfields. In finance, Markov matrices are used to model the rating migrationsof obligors. For instance, a 1-year rating transition matrix is simply a probabilisticrepresentation of the possible credit ratings an obligor could havein 1 year. The probability of migrating to another rating grade is a functionof the current credit rating of the obligor.